Problem 1 was: Find a vector equation for the line segment that joins $P(2,1,-3)$ to $Q(1,0,-5)$.

My solution was $\vec r(t)=\langle 2-t,1-t,-3-2t\rangle, \quad 0\le t\le 1$, but other solutions may also be correct. To see how to arrive at the solution, see the example in section 13.1 in the textbook.

I was looking especially to see that the range of $t$-values was limited in your quiz solution. Without this, an equation for the entire line rather than just the segment. The book's technique for this is to form the sum $t\langle2,1,-3\rangle+(1-t)\langle1,0,-5\rangle$. You can think of this as a way of "weighting" the two points with percentages… for example, if $t=.25$, then one can think of the equation as giving the point that is "25%" of the distance between $(2,1,-3)$ and $(1,0,-5)$. So the points in between $P$ and $Q$ arise as percentages between "0%" and "100%", hence the values $0\le t\le 1$.

If you just take the two points and come up with the equation for the line separately, you might come up with something like $\vec r(t)=\langle 2+t,1+t,-3+2t\rangle$. In this case, the values of $t$ describing the segment are between $-1$ and $0$. So you have to be careful, depending on how you come up with the equation. The easiest way to check that your solution works is to substitute $t=0$ and $t=1$ into your equation for the line… you should come up with the two endpoints of the segment.